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Properties

Label 15.5.52006947097...3296.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,2^{18}\cdot 3^{6}\cdot 19^{5}\cdot 59^{2}\cdot 56190247^{2}$
Root discriminant $176.90$
Ramified primes $2, 3, 19, 59, 56190247$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T74

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16317276, 22459748, -21923712, 11728688, -2322456, 622152, -256416, -77804, 29008, -2864, 4064, -622, -7, -29, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 29*x^13 - 7*x^12 - 622*x^11 + 4064*x^10 - 2864*x^9 + 29008*x^8 - 77804*x^7 - 256416*x^6 + 622152*x^5 - 2322456*x^4 + 11728688*x^3 - 21923712*x^2 + 22459748*x - 16317276)
 
gp: K = bnfinit(x^15 - x^14 - 29*x^13 - 7*x^12 - 622*x^11 + 4064*x^10 - 2864*x^9 + 29008*x^8 - 77804*x^7 - 256416*x^6 + 622152*x^5 - 2322456*x^4 + 11728688*x^3 - 21923712*x^2 + 22459748*x - 16317276, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 29 x^{13} - 7 x^{12} - 622 x^{11} + 4064 x^{10} - 2864 x^{9} + 29008 x^{8} - 77804 x^{7} - 256416 x^{6} + 622152 x^{5} - 2322456 x^{4} + 11728688 x^{3} - 21923712 x^{2} + 22459748 x - 16317276 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-5200694709710314084929511688503296=-\,2^{18}\cdot 3^{6}\cdot 19^{5}\cdot 59^{2}\cdot 56190247^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $176.90$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 19, 59, 56190247$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{65376165400972154656997928389185995686012484} a^{14} + \frac{409126351887277852372535222341954787843683}{65376165400972154656997928389185995686012484} a^{13} + \frac{4428261252325136164467097916349279568623033}{21792055133657384885665976129728665228670828} a^{12} + \frac{6725530122562675811929303684183036270788443}{65376165400972154656997928389185995686012484} a^{11} - \frac{2627885218360966600683799063702572284563061}{10896027566828692442832988064864332614335414} a^{10} - \frac{1606807671847658571381044777714603249792245}{16344041350243038664249482097296498921503121} a^{9} + \frac{3746437071821922435949373592725973893498125}{32688082700486077328498964194592997843006242} a^{8} + \frac{6036229903550286318420847654227126272562212}{16344041350243038664249482097296498921503121} a^{7} - \frac{12748507867645649211826781825866712676144629}{32688082700486077328498964194592997843006242} a^{6} - \frac{5146749658901222748970669563785739393006133}{32688082700486077328498964194592997843006242} a^{5} - \frac{4968321170333858107209816836233697225042848}{16344041350243038664249482097296498921503121} a^{4} - \frac{8143702824405115432294888877880868175687489}{16344041350243038664249482097296498921503121} a^{3} + \frac{4194080132913777542062279215089711594988160}{16344041350243038664249482097296498921503121} a^{2} + \frac{2580269661649088074234940093860524405909860}{16344041350243038664249482097296498921503121} a + \frac{413383140285402061475346005763484739914531}{5448013783414346221416494032432166307167707}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 741199299947 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T74:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 24000
The 40 conjugacy class representatives for [1/2.F(5)^3]S(3)
Character table for [1/2.F(5)^3]S(3) is not computed

Intermediate fields

3.1.76.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 30 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ R

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.16.26$x^{12} + 2 x^{11} + 2 x^{9} + 2 x^{7} + 2 x^{5} + 2 x^{4} + 2 x^{2} + 2$$12$$1$$16$12T63$[4/3, 4/3, 5/3, 5/3]_{3}^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$59$$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
59.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
59.4.2.2$x^{4} - 59 x^{2} + 6962$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
59.8.0.1$x^{8} - x + 14$$1$$8$$0$$C_8$$[\ ]^{8}$
56190247Data not computed